1. For example, at the end of the definitions of Book I:,"I use the words 'attraction', 'impulse', or 'tendency' of anything toward the center interchangeably with one another, considering these forces not physically but only mathematically. Whence the reader should beware lest he think that by such words I am anywhere defining a species or mode of action or a physical cause or explanation, or that I am actually and physically attributing forces to centers (which are mathematical points), if perhaps I should say either that centers attract or that centers have forces." Newton, 1687, pp. 4-5.

2. So it appeared to Joseph Needham viewing the origins of modern science from the perspective of his study of Chinese science; see Needham 1959, Sect. 19(k).

3. "Sed μηχανικην eam
Geometriae partem intelligimus, quae *Motum* tractat:
atque Geometricis rationibus, &
αποδεικτικως,
inquirit, quâ vi quisque motus peragatur." Wallis 1670, p. 1. Cf. Newton 1687, Praef. ad
lect.: "*Quo sensu *Mechanica rationalis* erit
Scientia Motuum qui ex viribus quibuscunque resultant, &
virium quae ad motus quoscunque requiruntur, accurate proposita
ac demonstrata*." On the varied and changing meanings of
"mechanics" in the 16th and 17th centuries, see Gabbey 1992.

4. Galileo 1969, Vol. I, p. 281.

5. Fontenelle 1725, p. 87.

6. Newton 1687, Book I, Sect. VI, Lemma XXVIII: "There is no oval figure, of which the area cut off by any straight line can be found generally by means of equations finite in the number of terms and dimensions."

7. See Leibniz 1686 and Leibniz 1694a. Cf. Bos 1988.

10. "It seems to me that the everyday
practice of the famous Arsenal of you gentlemen of Venice, in
particular in that part called 'mechanics', opens to speculative
minds a wide field for philosophizing." Galileo 1638, p. 1. Cf.
Robert Boyle's (1627-91) brief argument "That the Goods of
Mankind may be much Increased by the Naturalist's Insight into
Trades", resting on the propositions that "the
phaenomena afforded by the trades are (most of them) a part of
the history of nature, and therefore may both challenge the
naturalist's curiosity, and to [!] add to his knowledge,"
and that "[the phaenomena of trades] show us nature in
motion, and that too, when she is (as it were) put out of her
course, by the strength or skill of man, which I have formerly
noted to be the most instructive condition, wherein we can behold
her." (*Works*, III, 442-3; as published in Boyle 1965, pp. 163-5).

11. Kepler to Herwart von Hohenberg, 10 Feb. 1605, in Kepler 1937--, vol.. XV, p. 146.

12. Gabbey 1990. For a more extensive survey of the changing meaning of "mechanics" in the seventeenth century, see Gabbey 1992.

13. For a survey and analysis of various notions of perpetual motion and their relation to the principles of mechanics in the seventeenth century, see Gabbey 1985.

14. Stevin 1586, Book I, Prop. 19, pp. 175-179. One may question the cogency of Stevins' reasoning here, to wit, whether the perpetual motion he describes is in fact impossible in the abstract (see Gabbey 1985, 74, n.26). But the point at hand concerns his translation of its practical impossibility into a mathematical argument.

15. The frontispiece of Fontana's treatise on moving the obelisk graphically illustrates the problem of scaling, as many of the competing proposals portrayed there are clearly unfeasible when scaled to the full-sized obelisk.

16. Letter to Castelli, 21.XII.1613; in Galileo 1969, vol. I, p. 177.

17. See, for example, Chap. [14] of his *De
motu* [*antiquiora*] (*ca.* 1590) in Galileo 1960, p.64.

18. On Galileo's early work in mechanics, see Clavelin 1974, Chap. 3.

19. On the relation of Galileo's new science of motion to medieval sources as mediated by the sixteenth-century scholastic curriculum, see Wallace 1984, esp. Chap. 4.

20. On the "mean speed theorem" and medieval kinematics in general, see Clagett 1959, esp. Part II.

21. For an overview, see Clagett 1968.

23. Galileo 1638, p. 28.

24. Cavalieri to Galileo, 2 October 1634,
in Galileo 1890-1909,
vol. XVI, pp. 136-7; 19 December 1634, *Ibid.*, pp. 175-6.

25. On the general question of the concept
of *momentum* in Galileo's thought, see Galluzzi 1979.

26. Schuster 1977, pp. 299-352.

27. A *cycloid* is the path traced
by a fixed point on the circumference of a circle as the circle
rolls along a straight line; see below, §4.2.

28. For a general account of Huygens' mechanics see Gabbey 1980; for its role in his work on clocks and the determination of longitude, see Mahoney 1980.

29. Galileo had first called attention to
the pendulum as a system that would continue to oscillate
uniformly, falling and then rising to the same height, were it
not for external damping forces. He established that the period
is independent of the weight of the bob and asserted that it is
also independent of the amplitude. Although Galileo saw the
pendulum as a time-keeping device and suggested attaching it to
the escapement of a clock, he himself used it primarily as a
means of experimenting with falling bodies, using it to argue the
speed of fall is independent of weight and that a body in falling
acquires enough *momento* or *impeto* to raise it
to its original height. See Bedini
1991.

30. Torricelli, *De motu gravium
naturaliter descendentium et projectorum*, in Torricelli 1919, vol.
2, p.105. E.J. Dijksterhuis first brought out the relation
between Torricelli and Huygens; see Dijksterhuis 1961,
pp. 370-2.

31. Huygens 1659, Prop.XI.

32. Note, for example, the similarity between Huygens' remark in 1675 that "the quantity of incitation at each instant of motion is measured by the force required to prevent the body from starting to move at the place where it is and in the direction it is headed," (Huygens 1888, Vol. XVIII, p. 496) and Newton's definition of an impressed force as "the action carried out on a body to change its state either of motion or of moving uniformly in a straight line" (Newton 1687, Book I, def. 4). Cf. Gabbey 1980, pp. 176-77.

33. Herivel
1965, 129-130, from Newton's *Waste Book* in a section
dating from 1664. Newton recalled this earlier derivation in a
scholium to Proposition I,4 of the *Principia* (Newton
1687), where he emphasized the total force exerted by successive
impacts. Over a given period of time, he reasoned, that force is
proportional to the velocity and to the number of reflections.
For polygon of any given number of sides, the velocity will be as
the distance traveled, and the number of reflections will be as
that distance divided by the length of a side, which in turn is
proportional to the radius. Hence the total force will be as the
square of the distance divided by the radius, "and thus, if
the polygon with infinitely diminished sides coincides with the
circle, as the square of the arc described in the given time
divided by the radius." Again the transition from inscribed
polygons to the circle as limit rests on a property that is
seemingly independent of the number or size of the sides.

34. For greater detail and further references, see Mahoney 1993.

35. Newton 1730, p. 396.

36. See Heilbron 1993.

37. Descartes to Mersenne, 28.VIII.1638, in Descartes 1879-1913, II, 309.

38. *Ibid.*, 313: "One should
also note that curves described by rolling circles (*roulettes*)
are entirely mechanical lines and count among those I have
rejected from my *Geometry*; that is why it is no wonder
that their tangents are not found by the rules I have set out
there."

39. Roberval 1693, pp. 1-89.

40. "To resolve problems by motion", <October, 1666>, in Newton 1967, vol. I, pp. 400-48.

41. As a simple example, let *y*^{2}
= *x*^{3}. Adding moments, (*y* + *qo*)^{2}
= (*x* + *po*)^{3}, or *y*^{2}
+ 2*yqo* + *q*^{2}*o*^{2} = *x*^{3}
+ 3*x*^{2}*po* + 3*xp*^{2}*o*^{2}
+ *p*^{3}*o*^{3}. Deleting the
equal terms and dividing the others by *o* yields 2*yq*
+ *q*^{2}*o* = 3*x*^{2}*p*
+ 3*xp*^{2}*o* + *p*^{3}*o*^{2}.
Since *o* is infinitely small, so too are all terms
containing it, whence 2*yq* = 3*x*^{2}*p*,
or *y*^{2}(2*q*/*y*) = *x*^{3}(3*p*/*x*),
as the rule states.

42. Newton, "De analysi per
aequationes numero terminorum infinitas", in Newton 1967, vol. II,
pp. 206-47; cf. the fully developed (but long unpublished) tract
"On the Methods of Series and Fluxions" (1671), in Newton 1967, vol. III,
pp. 32-353. Despite claims made during the priority dispute with
Leibniz, Newton did not replace *p* and *q* with
the now familiar dot notation, or "pricked letters" and
, until 1691 (*Ibid*., 72, n. 86).

43. Leibniz 1855, vol. I, app. II; summarized in English with translated excerpts in Child 1920, 65ff.

44. For a discussion of the Aristotelean
passages (*Metaphysics* VI, 1,1026^{a}23-7, and
XI, 7, 1064^{b}8-9) and their subsequent interpretation,
see Sasaki 1988, Chap.6.
More generally, see Crapulli
1965.

45. *In artem analyticen isagoge*,
in Viète 1646, p. 1:
Est veritatis inquirendae via quaedam in Mathematicis, quam Plato
primus invenisse dicitur, à Theone nominata Analysis, & ab
eodem definita, Adsumptio quaesiti tanquam concessiper
consequentia ad verum concessum. Ut contrà Synthesis, Adsumptio
concessi per consequentia ad quaesiti finem &
comprehensionem.

46. The precise meaning of Pappus' description of analysis and synthesis is a matter of dispute. See Mahoney 1968, Hintikka and Remes 1974, and, most generally, Knorr 1993.

47. Mahoney 1994, Chap. 2, and Morse 1981.

48. The constituents of his *Opus
restitutae mathematicae analyseos, seu algebra nova*, all
composed in the 1570s and '80s, were published separately over
the course of several decades and brought together for the first
time in Viète 1646.
They include* In artem analyticem isagoge* (Tours, 1591), *Ad
logisticen speciosam notae priores* (Paris, 1631), *Zeteticorum
libri quinque* (Tours, 1593), *De aequationum recognitione
et emendatione tractatus duo* (Paris, 1615), *De numerosa
potestatum ad exegesin resolutione* (Paris, 1600), *Effectionum
geometricarum canonica recensio* (Tours, 1592), *Supplementum
geometriae* (Tours, 1593), *Theoremata ad sectiones
angulares* (Paris, 1615), and *Variorum de rebus
mathematicis responsorum liber VIII* (Tours, 1593). For the
complex history of Viète's works, see Van Egmond 1985, and,
more extensively, Grisard
1976.

49. To these, Tartaglia, Cardano, and Ferrari had added general solutions of the cubic and quartic equation. On the Arabic background and its influence on European developments, see Rashed 1984.

50. See, for example, Rule IV of the *Regulae
ad directionem ingenii*, in Descartes 1879-1913,
vol. X, p. 377: "Finally, there have been some most
ingenious men who have tried in this century to revive the same
[true mathematics]; for it seems to be nothing other than that
art which they call by the barbarous name of 'algebra', if only
it could be disentangled from the multiple numbers and
inexplicable figures that overwhelm it, so that it would no
longer lack the clarity and simplicity that we suppose should
obtain in a true mathematics."

51. In the sense that, given any two terms
of the progression, the smaller can be added to itself a
sufficient number of times to exceed the greater. Adding a line
to itself however many times will not produce an area. See his *Géométrie*
(Leiden, 1637, as part of the *Discours de la méthode*
and* Essais*; repr. separately with English trans. in Descartes 1954), 297-8.

52. Descartes 1954, Book
III, p.380: "Au reste tant les vrayes racines que les
fausses ne sont pas tousiours relles; mais quelquefois seulement
imaginaires; c'est à dire qu'on peut bien tousiours en imaginer
autant que iay dit en chasque Equation; mais qu'il n'y a
quelquefois aucune quantité, qui corresponde a celles qu'on
imagine. comme encore qu'on en puisse imaginer trois en celle cy,
*x*^{3} - 6*xx* + 13*x* - 10 = 0, il
n'y en a toutefois qu'une reelle, qui est 2, & pour les deux
autres, quoy qu'on les augmente, ou diminue, ou multipie en la
façon que ie viens d'expliquer, on ne sçauroit les rendres
autres qu'imaginaires." Just over a century later, Euler
commented on the analytic value of imaginaries: "We must
finally drop our concern that the doctrine of impossible numbers
might be viewed as useless fantasy. This concern is unfounded.
The doctrine of impossible numbers is in fact of the greatest
importance, since problems often arise in which one cannot know
immediately whether they demand something possible or impossible.
Whenever their solution leads to such impossible numbers, one has
a sure sign that the problem demands something impossible." Euler 1959 [1770], Pt. I,
sec. 1, par. 151.

53. Descartes 1954, Book II, p. 342.

54. For a detailed study of the development of Fermat's mathematics, see Mahoney 1994. With the exception of one treatise published anonymously in 1660, Fermat's works circulated only in manuscript copies during his lifetime. His son, Samuel, gathered a collection of them, which he issued as Fermat 1679. The main source today is Fermat 1891-1922.

55. It is, of course, one of the elementary symmetric functions of the equation. Viète's theory of equations brought out some, but not all, of these relations. Descartes focused on them as the links between an equation written as a polynomial and as a product of the linear binomials containing its roots.

56. Fermat first announced his method in an essay, "Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum", sent to Mersenne in 1636 but based on results achieved in 1629. The essay gave no hint of its origins, which Fermat revealed only in the course of a dispute with Descartes in 1638. For details, see Mahoney 1994, Chap. 4.

57. Viewing the origins of the calculus through Bishop Berkeley's lenses obscures the lines of thought that avoided infinites and infinitesimals altogether or that sought to keep them in a separate domain from finite quantities, linked only by common relationships.

58. Grosholz 1991 offers a sustained critique of this approach both to mathematics and to physics.

59. Cavalieri 1653; see Andersen 1985.

60. Fermat refined his methods of
quadrature in the late 1630s and early '40s, making them known
largely through his correspondence with Roberval; see Mahoney 1994, Chap. 5.
Roberval's methods similarly were known only to his network of
correspondents and perhaps also to his students at the Collège
royal de France (College de France after the Revolution) until
the publication of his *Traité des indivisibles* in Roberval 1693. For
Roberval's correspondence with Torricelli in the mid-1640s, on
which the discussion to follow is based, see Torricelli 1919, vol.
III, *passim*.

61. This is how the language of Cavalieri's method of indivisibles became attached to a method of infinitesimals conceptually different from it, thus creating the historical misunderstanding that the techniques of quadrature from which the integral calculus emerged stemmed from Cavalieri.

62. Fermat, *De aequationum localium
transmutatione et emendatione ad multimodam curvilineorum inter
se vel cum rectilineis comparationem ...*, [late 1650s], Fermat 1891-1922, vol.
I, pp. 255-85; see Mahoney
1994, Chap. 5, Sect. IV.

63. Barrow 1670; Gregory 1668; Mahoney 1990.

64. On Barrow, see Mahoney 1990.

65. Leibniz 1684; repub. in Leibniz 1849-63, vol. V, pp. 220-6.

66. Leibniz 1684, p. 223.

67. Leibniz 1686; Leibniz 1849-63, vol. V, pp. 226-33.

68. Leibniz to Huygens, 21 September 1691, Huygens 1888, Vol. X, p. 157.

69. Leibniz to Varignon, 2 February 1702, Leibniz 1849-63, vol.
IV, p. 92. "To tell the truth," he added a few months
later (*ibid.*, 110), "I am not altogether persuaded
myself that our infinites and infinitesimals should be considered
other than as ideal objects or as well founded fictions."
Cf. Leibniz to Johann Bernoulli, 29 July 1698, Leibniz 1849-63,
Vol. III, p. 524: Inter nos autem haec addo, quod et jam olim in
dicto Tractatu inedito adscripsi, dubitari posse an lineae rectae
infinitae longitudine et tamen terminatae revera dentur. Interim
sufficere pro Calculo, ut fingantur, uti imaginariae radices in
Algebra.

71. Leibniz to Varignon, 2 February 1702, Leibniz 1849-63, Vol. IV, pp. 93-4.

72. Newton 1687, p. 35. Cf. de Gandt 1986.

73. On the first strategy, see Jesseph 1989.

74. Cf. Breger 1990.

75. Bacon
1960 [1620], Book I, Aph. CXXIV, p. 114; where, however, *ipsissimae
res* is rendered "the very same things".

76. Fontenelle 1727,
Préface, [biv^{r}].

77. Barrow 1670, pp. 167-8.

78. Bacon, Nov. Org. I, Aph. 51.

79. Descartes, *Le monde**,
*Chap. 7, p.75. The final clause is quoted from the Bible, *Sapientia
*(*Wisdom*), VIII, 21.

80. The discussion in the *Two New
Sciences *of the paradoxes of the infinite may have made them
understandable to a wider, popular audience, but it did not go
beyond what Aristotle had said.

81. Even if, to preserve the continuity of
space identified with matter, Descartes was not prepared to
accept any minimal size of the particles of matter but rather
posited potentially infinite subdivision; see, for example, *Principles
of Philosophy*, II, 34.

82. Varignon takes *dt* as
constant, i.e. *ddt* = 0. Doing so in Leibnizian calculus
is equivalent in the modern form to choosing *t* as the
independent variable, i.e. as the variable with respect to which
one is differentiating. The Leibnizian version allows a greater
flexibility in analyzing differential relations, as Bernoulli had
shown in his lectures on the method of integrals and as Varignon
emphasized in his later memoirs. On the mathematical point, cf. Bos 1974.